To solve this problem, we can use the ideal gas law equation, which states:
PV = nRT
Where:
P is the absolute pressure
V is the volume
n is the number of moles
R is the ideal gas constant (8.314 J/mol·K)
T is the temperature
We are given:
Initial pressure, P1 = 4.69 x 10^5 Pa
Final pressure, P2 = 6.75 x 10^5 Pa
Volume, V = 4.1 x 10^-4 m^3
Temperature, T = 296 K
Using the ideal gas law, we can rearrange the equation to solve for the number of moles (n):
n = PV / RT
First, we need to convert the pressure from pascals (Pa) to atmospheres (atm) since the ideal gas constant (R) is typically expressed in those units. 1 atm = 101325 Pa.
P1 = 4.69 x 10^5 Pa = 4.61 atm
P2 = 6.75 x 10^5 Pa = 6.66 atm
Now, we can plug in the values into the equation:
n = (P2 * V) / (R * T)
n = (6.66 atm * 4.1 x 10^-4 m^3) / (8.314 J/mol·K * 296 K)
Simplifying the equation:
n = 0.0187 mol
Therefore, you would need to pump approximately 0.0187 moles of air into the tire to increase the pressure from 4.69 x 10^5 Pa to 6.75 x 10^5 Pa, keeping the temperature and volume constant.